Games Of Strategy Chapter 4 Dixit, Skeath, and Reiley Simultaneous Move Games: Continuous Strategies, Discussion, and Evidence Games Of Strategy Chapter 4 Dixit, Skeath, and Reiley

Terms to Know Best-response curve Quantal response equilibrium Best-response rule Continuous strategy Never a best response Quantal response equilibrium Rationalizability Rationalizable Refinement

Introductory Game The professor will have you count off All individuals that have an even number will be on one side of the room and be known as the x players while the odds will be on the other side of the room and known as the y players Suppose that each of you would like to maximize your profits where  = P*Q – cQ where P represents your price, Q represents the quantity demanded of your product, and c represents the unit cost of Q

Introductory Game Cont. The quantity demanded for product x can be defined as Qx = 100 – 7Px + Py where Px is the price player x chooses and Py is the price player y chooses The quantity demanded for product y can be defined as Qy = 100 – 7Py + Px where Py is the price player y chooses and Px is the price player x chooses The cost of production for each player is 8

Introductory Game Cont. Each side is allowed to work as a group On a piece of paper write the names of your teammates and the price your team will submit

Games Discussion How did your team figure out its best response to the other team?

Continuous Strategy Games There are games of strategy where the strategies that can be chosen are not necessarily discrete actions This can make it difficult and challenging to try to represent the game in a matrix form Even if the decisions are discrete in nature, e.g., price competition, there may be too many to deal with in a matrix Hence we need to figure out a way to analyze games when the decisions are continuous or numerous

Price Competition Example 1 Suppose we have two players x and y whose each have a goal of maximizing profit The quantity demanded for product x can be defined as Qx = 44 – 4Px + 2Py where Px is the price player x chooses and Py is the price player y chooses The quantity demanded for product y can be defined as Qy = 100 – 4Py + 2Px where Py is the price player y chooses and Px is the price player x chooses

Price Competition Example 1 Cont. Assume that the cost for each individual is $1 for each output Q produced Step 1: Set-up and solve each players profit max problem Step 2: Find each players best response function Step 3: Solve simultaneously the two players best response function to find each’s equilibrium

Price Competition Example 1 Cont. Step 4: Calculate the resulting output and profit at the equilibrium solution Solution will be done in class

Visual Representation of the Solution for Example 1 Py BR1(Py) BR2(Px) 8 6 6 8 Px

Collusion An agreement between rivals to work together Collusion can occur when there are a small number of players in a game When there are a small number of sellers, then it is said that there is an oligopoly; two sellers is known as a duopoly

Collusion Cont. When there are a small number of buyers, then it is said that there is an oligoposony A cartel is an organization that tries to collude to raise prices

What If The Two Competitors Could Collude in Example 1 Suppose that player x and y decided that they were going to charge the same price through collusion This would imply that Qx = 44 – 4P + 2P and Qy = 44 – 4P + 2P each with a cost of 1 What would be the optimal price?

Price Competition Example 2 Suppose we have two players x and y whose each have a goal of maximizing profit The quantity demanded for product x can be defined as Qx = 42 – 6Px + 6Py where Px is the price player x chooses and Py is the price player y chooses The quantity demanded for product y can be defined as Qy = 42 – 3Py + 6Px where Py is the price player y chooses and Px is the price player x chooses

Price Competition Example 2 Cont. Assume that the cost for player x is $3 and the cost for player y is $6 for each output Q produced Solution done in class Draw the problem graphically

Cournot Quantity Competition Suppose there were two companies that competed based on quantity Assume the demand curve for the product can be represented as P = 100 – 2Q where Q = qx +qy, where qx represents the amount that x would produce and qy represents the amount y would produce Suppose each face a cost of $20 for each unit each produces

Cournot Quantity Competition What is each competitors profit function? What is each players best response function in terms of quantity? What is the equilibrium quantity that would be chosen by each competitor? What is the profit of each? What would be the profit if they could collude?

Bertrand Price Competition Assume the same demand function and cost from the Cournot game from before Instead of the two players competing on quantity, assume that they compete on price Further assume that if they each have the same price, then they will split the market evenly Otherwise, the competitor with the lower price gets all the consumers What is the equilibrium to this game?

Criticism of Nash It does not take into account risk very well There can be multiple Nash equilibria The Nash equilibrium requires rationality How did the textbook authors counter these claims?

Never a Best Response and Rationalizable This property is stronger in terms of eliminating a strategy than dominance of a strategy You may be able to eliminate a strategy even though it is not dominated by using the argumentation of never a best response Strategies are said to be rationalizable if they survive the property of never a best response

Never a Best Response and Rationalizable Cont. If a game has a Nash, then it is rationalizable Even if there are no Nash, there can be rationalizable outcomes Rationalizability can take us to a Nash Equilibrium

Example of Never a Best Response Player 2 P2S1 P2S2 P2S3 P2S4 Player 1 P1S1 1 , 8 3 , 6 8 , 1 1 , 2 P1S2 6 , 3 4 , 4 6 , 2 P1S3 P1S4 1 , 1 1 , -1 11 , 0

Final Discussion, Questions, and Thoughts